Subgroups — C₅₃×#19640 #27227152

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3206	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3141	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1718	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1717	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1711	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1708	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#532	C₂²	1	1	yes	characteristic
13	C₅₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#500	C₂²	1	1	yes	characteristic
17	C₅₃×#498	C₂²	1	2	yes	—
18	C₅₃×#493	C₂²	1	1	yes	characteristic
19	C₅₃×#492	C₂²	1	2	yes	—
20	C₅₃×#489	—	2	2	no	—
21	C₅₃×#489	—	2	2	no	—
22	C₅₃×#487	—	2	2	no	—
23	C₅₃×#487	C₂²	1	2	yes	—
24	C₅₃×#487	C₂²	1	2	yes	—
25	C₅₃×#487	C₂²	1	2	yes	—
26	C₅₃×#487	C₂²	1	2	yes	—
27	C₅₃×#487	C₂²	1	2	yes	—
28	C₅₃×#486	C₂²	1	2	yes	—
29	C₅₃×#486	C₂²	1	1	yes	characteristic
30	C₅₃×#486	C₂²	1	2	yes	—
31	C₅₃×#486	C₂²	1	1	yes	characteristic
32	C₅₃×#486	C₂²	1	2	yes	—
33	C₅₃×#481	—	2	2	no	—
34	C₅₃×#476	—	2	2	no	—
35	C₅₃×#466	—	2	2	no	—
36	C₅₃×#461	—	2	2	no	—
37	C₅₃×#448	—	2	2	no	—
38	C₄×D₈×C₅₃	—	2	2	no	—
39	C₅₃×#431	C₂²	1	1	yes	characteristic
40	C₅₃×#417	—	2	2	no	—
41	C₅₃×#417	—	2	2	no	—
42	C₂×#103×C₅₃	—	2	2	no	—
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	—
46	C₈²×C₅₃	C₂²	1	2	yes	—
47	C₂×#41×C₅₃	—	2	1	no	—
48	C₂×#41×C₅₃	—	2	1	no	—
49	C₂²×Q₈×C₅₃	—	2	1	no	—
50	C₂²×Q₈×C₅₃	—	2	1	no	—
51	C₂×SD₁₆×C₅₃	—	2	4	no	—
52	C₂×D₈×C₅₃	—	2	2	no	—
53	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
54	C₂²×C₈×C₅₃	D₄	1	2	yes	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	—	2	1	no	—
59	C₄⋊Q₈×C₅₃	—	2	1	no	—
60	C₄⋊Q₈×C₅₃	—	2	1	no	—
61	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
65	#127×C₅₃	C₂³	1	1	yes	characteristic
66	#127×C₅₃	C₂³	1	1	yes	characteristic
67	#124×C₅₃	—	2	1	no	—
68	#124×C₅₃	—	2	1	no	—
69	#121×C₅₃	—	2	1	no	—
70	#121×C₅₃	—	2	1	no	—
71	#121×C₅₃	—	2	1	no	—
72	#121×C₅₃	—	2	1	no	—
73	C₄×D₄×C₅₃	—	2	1	no	—
74	C₄×D₄×C₅₃	—	2	1	no	—
75	C₄×D₄×C₅₃	—	2	1	no	—
76	C₄×D₄×C₅₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	#107×C₅₃	—	2	2	no	—
83	#106×C₅₃	—	2	4	no	—
84	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
85	#103×C₅₃	—	2	4	no	—
86	#103×C₅₃	—	2	4	no	—
87	#103×C₅₃	—	2	4	no	—
88	#103×C₅₃	—	2	4	no	—
89	#103×C₅₃	—	2	4	no	—
90	#102×C₅₃	—	2	4	no	—
91	#102×C₅₃	—	2	4	no	—
92	#102×C₅₃	—	2	2	no	—
93	#102×C₅₃	—	2	2	no	—
94	#102×C₅₃	—	2	4	no	—
95	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
96	#97×C₅₃	D₄	1	2	yes	—
97	C₄×C₈×C₅₃	D₄	1	2	yes	—
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₅₃	C₂³	1	2	yes	—
100	C₄×C₈×C₅₃	C₂³	1	2	yes	—
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	#41×C₅₃	—	4	4	no	—
103	#41×C₅₃	—	4	4	no	—
104	C₂×Q₈×C₅₃	—	2	1	no	—
105	C₂×Q₈×C₅₃	—	2	1	no	—
106	C₂×Q₈×C₅₃	—	2	2	no	—
107	C₂×Q₈×C₅₃	—	2	2	no	—
108	C₂×Q₈×C₅₃	—	4	4	no	—
109	C₂×Q₈×C₅₃	—	4	1	no	—
110	C₂×Q₈×C₅₃	—	4	1	no	—
111	C₂×Q₈×C₅₃	—	8	2	no	—
112	C₂×D₄×C₅₃	—	2	1	no	—
113	C₂×D₄×C₅₃	—	2	1	no	—
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	1	no	—
117	C₂×D₄×C₅₃	—	2	1	no	—
118	C₂×D₄×C₅₃	—	4	1	no	—
119	C₂×D₄×C₅₃	—	4	1	no	—
120	C₂²×C₄×C₅₃	—	4	1	no	—
121	C₂²×C₄×C₅₃	—	4	1	no	—
122	C₂²×C₄×C₅₃	—	4	1	no	—
123	C₂²×C₄×C₅₃	—	4	1	no	—
124	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
127	SD₁₆×C₅₃	—	8	4	no	—
128	D₈×C₅₃	—	4	4	no	—
129	M₁₆×C₅₃	—	2	4	no	—
130	C₂×C₈×C₅₃	—	4	2	no	—
131	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
132	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
134	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	—	2	2	no	—
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	—	4	1	no	—
144	C₄⋊C₄×C₅₃	—	4	1	no	—
145	C₄⋊C₄×C₅₃	—	4	1	no	—
146	C₄⋊C₄×C₅₃	—	4	1	no	—
147	C₂²⋊C₄×C₅₃	—	4	1	no	—
148	C₂²⋊C₄×C₅₃	—	4	1	no	—
149	C₂²⋊C₄×C₅₃	—	4	1	no	—
150	C₂²⋊C₄×C₅₃	—	4	1	no	—
151	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
155	C₂³×C₅₃	—	4	1	no	—
156	C₂³×C₅₃	—	4	1	no	—
157	C₂³×C₅₃	#127	1	1	yes	characteristic
158	Q₈×C₅₃	—	4	2	no	—
159	Q₈×C₅₃	—	4	2	no	—
160	Q₈×C₅₃	—	4	4	no	—
161	Q₈×C₅₃	—	4	4	no	—
162	Q₈×C₅₃	—	8	2	no	—
163	Q₈×C₅₃	—	8	2	no	—
164	D₄×C₅₃	—	4	2	no	—
165	D₄×C₅₃	—	4	2	no	—
166	D₄×C₅₃	—	4	2	no	—
167	D₄×C₅₃	—	4	2	no	—
168	D₄×C₅₃	—	4	2	no	—
169	D₄×C₅₃	—	4	2	no	—
170	D₄×C₅₃	—	8	2	no	—
171	D₄×C₅₃	—	8	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	4	1	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	—	4	2	no	—
178	C₂×C₄×C₅₃	—	8	2	no	—
179	C₂×C₄×C₅₃	—	8	2	no	—
180	C₂×C₄×C₅₃	—	8	2	no	—
181	C₂×C₄×C₅₃	—	8	2	no	—
182	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
183	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
184	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
185	C₂×C₄×C₅₃	#135	1	4	yes	—
186	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
189	C₈×C₅₃	—	2	4	no	—
190	C₈×C₅₃	—	2	2	no	—
191	C₈×C₅₃	—	2	2	no	—
192	C₈×C₅₃	—	2	4	no	—
193	#19640	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂²×C₅₃	—	2	1	no	—
195	C₂²×C₅₃	—	2	1	no	—
196	C₂²×C₅₃	—	8	1	no	—
197	C₂²×C₅₃	—	8	1	no	—
198	C₂²×C₅₃	—	8	1	no	—
199	C₂²×C₅₃	—	8	1	no	—
200	C₂²×C₅₃	—	8	1	no	—
201	C₂²×C₅₃	—	8	1	no	—
202	C₂²×C₅₃	#495	1	2	yes	—
203	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
204	C₄×C₅₃	—	2	1	no	—
205	C₄×C₅₃	—	2	1	no	—
206	C₄×C₅₃	—	8	1	no	—
207	C₄×C₅₃	—	8	1	no	—
208	C₄×C₅₃	—	8	2	no	—
209	C₄×C₅₃	—	8	2	no	—
210	C₄×C₅₃	#495	1	2	yes	—
211	C₄×C₅₃	C₂×#135	1	2	yes	—
212	C₄×C₅₃	#575	1	2	yes	—
213	C₄×C₅₃	#577	1	2	yes	—
214	#3227	C₂×C₅₃	1	1	yes	characteristic, complemented
215	#3206	C₂×C₅₃	1	1	yes	characteristic, complemented
216	#3169	C₂×C₅₃	1	1	yes	characteristic
217	#3141	C₂×C₅₃	1	1	yes	characteristic, complemented
218	#3141	C₂×C₅₃	1	1	yes	characteristic, complemented
219	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
220	#1718	C₂×C₅₃	1	2	yes	complemented
221	#1717	C₂×C₅₃	1	2	yes	complemented
222	#1711	C₂×C₅₃	1	2	yes	complemented
223	#1708	C₂×C₅₃	1	2	yes	complemented
224	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
225	C₂×C₅₃	—	2	2	no	—
226	C₂×C₅₃	—	16	1	no	—
227	C₂×C₅₃	—	16	1	no	—
228	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
229	C₂×C₅₃	#3153	1	1	yes	characteristic
230	C₂×C₅₃	#3154	1	1	yes	characteristic
231	#532	C₂²×C₅₃	1	1	yes	characteristic
232	#532	C₂²×C₅₃	1	1	yes	characteristic
233	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
234	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
235	#500	C₂²×C₅₃	1	1	yes	characteristic
236	#498	C₂²×C₅₃	1	2	yes	—
237	#493	C₂²×C₅₃	1	1	yes	characteristic
238	#492	C₂²×C₅₃	1	2	yes	—
239	#489	—	2	2	no	—
240	#489	—	2	2	no	—
241	#487	—	2	2	no	—
242	#487	C₂²×C₅₃	1	2	yes	—
243	#487	C₂²×C₅₃	1	2	yes	—
244	#487	C₂²×C₅₃	1	2	yes	—
245	#487	C₂²×C₅₃	1	2	yes	—
246	#487	C₂²×C₅₃	1	2	yes	—
247	#486	C₂²×C₅₃	1	2	yes	—
248	#486	C₂²×C₅₃	1	2	yes	—
249	#486	C₂²×C₅₃	1	1	yes	characteristic
250	#486	C₂²×C₅₃	1	1	yes	characteristic
251	#486	C₂²×C₅₃	1	2	yes	—
252	#481	—	2	2	no	—
253	#476	—	2	2	no	—
254	#466	—	2	2	no	—
255	#461	—	2	2	no	—
256	#448	—	2	2	no	—
257	C₄×D₈	—	2	2	no	—
258	#431	C₂²×C₅₃	1	1	yes	characteristic
259	#417	—	2	2	no	—
260	#417	—	2	2	no	—
261	C₂×#103	—	2	2	no	—
262	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
264	#322	C₂²×C₅₃	1	2	yes	—
265	C₈²	C₂²×C₅₃	1	2	yes	—
266	C₅₃	#19640	1	1	yes	characteristic, direct product factor, Sylow
267	C₂×#41	—	2	1	no	—
268	C₂×#41	—	2	1	no	—
269	C₂²×Q₈	—	2	1	no	—
270	C₂²×Q₈	—	2	1	no	—
271	C₂×SD₁₆	—	2	4	no	—
272	C₂×D₈	—	2	2	no	—
273	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
274	C₂²×C₈	D₄×C₅₃	1	2	yes	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	—	2	1	no	—
278	C₄⋊Q₈	—	2	1	no	—
279	C₄⋊Q₈	—	2	1	no	—
280	C₄⋊Q₈	—	2	1	no	—
281	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
282	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
283	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
284	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
285	#127	C₂³×C₅₃	1	1	yes	characteristic
286	#127	C₂³×C₅₃	1	1	yes	characteristic
287	#124	—	2	1	no	—
288	#124	—	2	1	no	—
289	#121	—	2	1	no	—
290	#121	—	2	1	no	—
291	#121	—	2	1	no	—
292	#121	—	2	1	no	—
293	C₄×D₄	—	2	1	no	—
294	C₄×D₄	—	2	1	no	—
295	C₄×D₄	—	2	1	no	—
296	C₄×D₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
302	#107	—	2	2	no	—
303	#106	—	2	4	no	—
304	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
305	#103	—	2	4	no	—
306	#103	—	2	4	no	—
307	#103	—	2	4	no	—
308	#103	—	2	4	no	—
309	#103	—	2	4	no	—
310	#102	—	2	4	no	—
311	#102	—	2	4	no	—
312	#102	—	2	2	no	—
313	#102	—	2	2	no	—
314	#102	—	2	4	no	—
315	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
316	#97	D₄×C₅₃	1	2	yes	—
317	C₄×C₈	D₄×C₅₃	1	2	yes	—
318	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
319	C₄×C₈	C₂³×C₅₃	1	2	yes	—
320	C₄×C₈	C₂³×C₅₃	1	2	yes	—
321	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
322	#41	—	4	4	no	—
323	#41	—	4	4	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	4	4	no	—
329	C₂×Q₈	—	4	1	no	—
330	C₂×Q₈	—	4	1	no	—
331	C₂×Q₈	—	8	2	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	4	1	no	—
339	C₂×D₄	—	4	1	no	—
340	C₂²×C₄	—	4	1	no	—
341	C₂²×C₄	—	4	1	no	—
342	C₂²×C₄	—	4	1	no	—
343	C₂²×C₄	—	4	1	no	—
344	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
347	SD₁₆	—	8	4	no	—
348	D₈	—	4	4	no	—
349	M₁₆	—	2	4	no	—
350	C₂×C₈	—	4	2	no	—
351	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
352	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
353	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
354	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	4	1	no	—
364	C₄⋊C₄	—	4	1	no	—
365	C₄⋊C₄	—	4	1	no	—
366	C₄⋊C₄	—	4	1	no	—
367	C₂²⋊C₄	—	4	1	no	—
368	C₂²⋊C₄	—	4	1	no	—
369	C₂²⋊C₄	—	4	1	no	—
370	C₂²⋊C₄	—	4	1	no	—
371	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
372	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
373	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
374	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
375	C₂³	—	4	1	no	—
376	C₂³	—	4	1	no	—
377	C₂³	#127×C₅₃	1	1	yes	characteristic
378	Q₈	—	4	2	no	—
379	Q₈	—	4	2	no	—
380	Q₈	—	4	4	no	—
381	Q₈	—	4	4	no	—
382	Q₈	—	8	2	no	—
383	Q₈	—	8	2	no	—
384	D₄	—	4	2	no	—
385	D₄	—	4	2	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	2	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	8	2	no	—
391	D₄	—	8	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	8	2	no	—
399	C₂×C₄	—	8	2	no	—
400	C₂×C₄	—	8	2	no	—
401	C₂×C₄	—	8	2	no	—
402	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
403	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
404	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
405	C₂×C₄	#135×C₅₃	1	4	yes	—
406	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
407	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
408	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
409	C₈	—	2	4	no	—
410	C₈	—	2	2	no	—
411	C₈	—	2	2	no	—
412	C₈	—	2	4	no	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	8	1	no	—
416	C₂²	—	8	1	no	—
417	C₂²	—	8	1	no	—
418	C₂²	—	8	1	no	—
419	C₂²	—	8	1	no	—
420	C₂²	—	8	1	no	—
421	C₂²	C₅₃×#495	1	2	yes	—
422	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
423	C₄	—	2	1	no	—
424	C₄	—	2	1	no	—
425	C₄	—	8	1	no	—
426	C₄	—	8	1	no	—
427	C₄	—	8	2	no	—
428	C₄	—	8	2	no	—
429	C₄	C₅₃×#495	1	2	yes	—
430	C₄	C₂×#135×C₅₃	1	2	yes	—
431	C₄	C₅₃×#575	1	2	yes	—
432	C₄	C₅₃×#577	1	2	yes	—
433	C₂	—	2	2	no	—
434	C₂	—	16	1	no	—
435	C₂	—	16	1	no	—
436	C₂	C₂×C₅₃×#495	1	1	yes	characteristic
437	C₂	C₅₃×#3153	1	1	yes	characteristic
438	C₂	C₅₃×#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅₃×#3227	C₂	semidirect
2. C₅₃×#3206	C₂	semidirect
4. C₅₃×#3141	C₂	semidirect
5. C₅₃×#3141	C₂	semidirect
6. C₂×C₅₃×#487	C₂	semidirect
7. C₅₃×#1718	C₂	semidirect
8. C₅₃×#1717	C₂	semidirect
9. C₅₃×#1711	C₂	semidirect
10. C₅₃×#1708	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
193. #19640	C₅₃	direct (N × H)
214. #3227	C₂×C₅₃	semidirect
215. #3206	C₂×C₅₃	semidirect
217. #3141	C₂×C₅₃	semidirect
218. #3141	C₂×C₅₃	semidirect
219. C₂×#487	C₂×C₅₃	semidirect
220. #1718	C₂×C₅₃	semidirect
221. #1717	C₂×C₅₃	semidirect
222. #1711	C₂×C₅₃	semidirect
223. #1708	C₂×C₅₃	semidirect
224. C₈×M₁₆	C₂×C₅₃	semidirect
266. C₅₃	#19640	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#3227	2
C₅₃×#3206	2
C₅₃×#3169	2
C₅₃×#3141	2
C₂×C₅₃×#487	2
C₅₃×#1718	2
C₅₃×#1717	2
C₅₃×#1711	2
C₅₃×#1708	2
C₈×M₁₆×C₅₃	2
#19640	53

Subgroups — C53×#19640 #27227152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19640 #27227152